Asymptotically Minimax Bayes Predictive Densities

Given a random sample from a distribution with density function that depends on an unknown parameter ., we are interested in accurately estimating the true parametric density function at a future observation from the same distribution. The asymptotic risk of Bayes predictive density estimates with Kullback-Leibler loss function $D(f_{\theta}\|\hat{f})=\int f_{\theta}\ {\rm log}\ (f_{\theta}/\hat{f})$ is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax. A result resembling Stein's paradox for estimating normal means by maximum likelihood holds for the uniform prior in the multivariate location family case: when the dimensionality of the model is at least three, the Jeffreys prior is minimax, though inadmissible. The Jeffreys prior is both admissible and minimax for one- and two-dimensional location problems.